The main motivation for the inequality (REF ) is establishing a Weyl's law for Schrödinger operators \(\hbar ^2\Delta +V\)  with a non-smooth potential \(V\)  under the semi-classical limit \(h\rightarrow 0\)  (see [1]}, [2]}, [3]}). As it turns out, the inequality (REF ) was already established by Rozenblum [4]}, [5]}. Another independent proof was provided by Lieb [6]}, [7]} (see [8]}, [9]}, [10]}, [11]}, [12]} for further alternative proofs). The equality (REF ) is now known as the Cwikel-Lieb-Rozenblum (CLR) inequality.
